
TL;DR
This paper introduces two new families of graphs, $G_m$ and $H_m$, generalizing Gr"otzsch graphs, with properties like non-planarity, triangle-freeness, Hamiltonicity, and specific chromatic numbers, expanding understanding of triangle-free graphs.
Contribution
The paper constructs and analyzes two new families of graphs, $G_m$ and $H_m$, generalizing Gr"otzsch graphs with detailed properties and comparisons to existing constructions.
Findings
$G_m$ is 4-chromatic and non-planar.
$H_m$ is 3-chromatic and non-planar.
Both families are triangle-free and Hamiltonian.
Abstract
The aim of this paper is to present a generalization of Gr\"otzsch graph. Inspired by structure of the Gr\"otzsch's graph, we present constructions of two families of graphs, and for odd and even values of respectively and on vertices. We show that each member of this family is non-planar, triangle-free, and Hamiltonian. Further, when is odd the graph is maximal triangle-free, and when is even, the addition of exactly edges makes the graph maximal triangle-free. We show that is 4-chromatic and is 3-chromatic for all . Further, we note some other properties of these graphs and compare with Mycielski's construction.
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Taxonomy
Topicsgraph theory and CDMA systems
